I've fixed more than my fair share of tire flats in my life, but that doesn't mean I can't learn anything new, quite the opposite. I had to take care of another this morning, and
Sheldon Brown's website (which, amazingly, is still around) provides a host of information about this - and many, many other things.
One thing I learned from Sheldon is to start, when removing the tire from the rim, near (but not right at) the valve with the tire levers; and, conversely, to push the final piece of tire back onto the rim near the valve when reinstalling the tire. Turns out this makes a substantial difference.
Outside of that I also learned today that patches apparently become useless as time goes by. When patching the first hole, the first three patches did not stick at all, even though I was doing everything right (even after double-checking what Sheldon had to say); when I used a different patch that I knew was newer, it worked beautifully.
Same for the next four patches I had to apply. (Don't ask; it was one of those days.) Here I used newer patches right away, and they all stuck beautifully.
One reason I have older patches is that several years ago, I got a big pack (100 of them), reasoning that I'd need them continually and that I could save a fair bit of money by buying them in bulk. It was a good idea in theory.
Finally - thank you for your website, Sheldon, and enjoy your rides, wherever you are now.